We provide local bounds for positive viscosity sub and supersolutions to a class of doubly nonlinear parabolic equations, <Equation ID="Equ77"> <EquationSource Format="TEX">$\begin{aligned} H(Du,D^2u)-u^\alpha u_t=0,\;\;0\le \alpha \le k-1,\;k> 1,\quad \hbox {in } \Omega \times [0,T), \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mi>u</mi> <mo>,</mo> <msup> <mi>D</mi> <mn>2</mn> </msup> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msup> <mi>u</mi> <mi>α</mi> </msup> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.277778em" /> <mspace width="0.277778em" /> <mn>0</mn> <mo>≤</mo> <mi>α</mi> <mo>≤</mo> <mi>k</mi> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.277778em" /> <mi>k</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <mtext>in</mtext> <mspace width="0.333333em" /> <mi mathvariant="normal">Ω</mi> <mo>×</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> is a bounded domain and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0<T\le \infty .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>T</mi> <mo>≤</mo> <mi>∞</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The spatial operator H is homogeneous with power k.

Local estimates for solutions of a class of nonlinear degenerate parabolic equations

期刊论文
发表日期：

2026年2月14日

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Tilak Bhattacharya,
Leonardo Marazzi

摘要

We provide local bounds for positive viscosity sub and supersolutions to a class of doubly nonlinear parabolic equations, $\begin{aligned} H(Du,D^2u)-u^\alpha u_t=0,\;\;0\le \alpha \le k-1,\;k> 1,\quad \hbox {in } \Omega \times [0,T), \end{aligned}$ $\begin{matrix} H (D u, D^{2} u) - u^{α} u_{t} = 0, 0 \leq α \leq k - 1, k > 1, in Ω \times [0, T), \end{matrix}$ where $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ is a bounded domain and $0<T\le \infty .$ $0 < T \leq \infty .$ The spatial operator H is homogeneous with power k.